Convergence of the CBHD Series and Associativity of the CBHD Operation

Abstract

THE aim of this chapter is twofold. On the one hand, we aim to study the 4 convergence of the Dynkin series

$$ \begin{array}{*{20}c} {uv: = } & {\sum\limits_{j = 1}^\infty {\left( {\sum\limits_{n = 1}^j {\frac{{\left( { - 1} \right)^{n + 1} }}{n}\,\sum\limits_{\begin{array}{*{20}c} {(h_1,k_1 ), \cdots (h_n,k_n ) \ne (0,0)} \\ {h_1 + k_1 + \cdots + h_n + k_n = j} \\ \end{array}} \times \frac{{(ad\,u)^{h_1 } (ad\,\upsilon )^{k_1 } \cdots (ad\,u)^{h_n } (ad\,\upsilon )^{k_n - 1} (\upsilon )}}{{h_1 ! \cdots h_n !k_1 ! \cdots k_n !(\sum\nolimits_{i = 1}^n {(h_i + k_i )} )}}} } \right),} } \\ \end{array} $$

in various contexts. For instance, this series can be investigated in any nilpotent Lie algebra (over a field of characteristic zero) where it is actually a finite sum, or in any finite dimensional real or complex Lie algebra and, more generally, its convergence can be studied in any normed Banach-Lie algebra (over R or C). For example, the case of the normed Banach algebras (becoming normed Banach-Lie algebras if equipped with the associated commutator) will be extensively considered here.